Ed(6ddlmZddlmZddlmZddlmZddlm Z m Z m Z m Z m Z ddlmZddlmZddlZd Zd Zd Zd Zd ZdZdZdZdZdZdZd*dZdZdZ dZ!dZ"dZ#dZ$dZ%dZ&dZ'd+dZ(d Z)d!Z*d"Z+d#Z,d$Z-d%Z.d&Z/d'Z0d(Z1d)Z2dS),)Dummy) nextprimecrt)PolynomialRing)gf_gcd gf_from_dictgf_gcdexgf_divgf_lcm)ModularGCDFailed)sqrtNc |j}|s|s|j|j|jfS|s5|j|jjkr| |j|j fS||j|jfS|s5|j|jjkr| |j |jfS||j|jfSdS)zn Compute the GCD of two polynomials in trivial cases, i.e. when one or both polynomials are zero. N)ringzeroLCdomainone)fgrs 6lib/python3.11/site-packages/sympy/polys/modulargcd.py _trivial_gcdr s 6D * *y$)TY..  * 4$+" " *2ty48)+ +di) ) * 4$+" " *2y$)+ +dh ) ) 4c|jj}|r|}|}||j|} |}||krnK||||z f||jzz |}f|}|}||||j||S)zM Compute the GCD of two univariate polynomials in `\mathbb{Z}_p[x]`. )rrdegreeinvertr mul_monom mul_ground trunc_ground)fpgppdomremdeglcinvdegrems r_gf_gcdr(#s '.C iikk 25!$$ cZZ\\F| v|o66AA%#&.QQQ__`abbC  c    ==BE1-- . . ; ;A > >>rc\|jj|j|j}d}t |}||zdkrt |}||zdk||}||}t |||}|}|S)a Compute an upper bound for the degree of the GCD of two univariate integer polynomials `f` and `g`. The function chooses a suitable prime `p` and computes the GCD of `f` and `g` in `\mathbb{Z}_p[x]`. The choice of `p` guarantees that the degree in `\mathbb{Z}_p[x]` is greater than or equal to the degree in `\mathbb{Z}[x]`. Parameters ========== f : PolyElement univariate integer polynomial g : PolyElement univariate integer polynomial r)rrgcdrrrr(r)rrgammar"r r!hpdeghps r_degree_bound_univariater/:s& FM  adAD ) )E A! A !)q. aLL !)q.   B   B R  B IIKKE Lrc T|}|jjd}|jj}t |dzD]N}t ||g|||z|||zgdd||f<O||S)a Construct a polynomial `h_{pq}` in `\mathbb{Z}_{p q}[x]` such that .. math :: h_{pq} = h_p \; \mathrm{mod} \, p h_{pq} = h_q \; \mathrm{mod} \, q for relatively prime integers `p` and `q` and polynomials `h_p` and `h_q` in `\mathbb{Z}_p[x]` and `\mathbb{Z}_q[x]` respectively. The coefficients of the polynomial `h_{pq}` are computed with the Chinese Remainder Theorem. The symmetric representation in `\mathbb{Z}_p[x]`, `\mathbb{Z}_q[x]` and `\mathbb{Z}_{p q}[x]` is used. It is assumed that `h_p` and `h_q` have the same degree. Parameters ========== hp : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_p` hq : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_q` p : Integer modulus of `h_p`, relatively prime to `q` q : Integer modulus of `h_q`, relatively prime to `p` Examples ======== >>> from sympy.polys.modulargcd import _chinese_remainder_reconstruction_univariate >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> p = 3 >>> q = 5 >>> hp = -x**3 - 1 >>> hq = 2*x**3 - 2*x**2 + x >>> hpq = _chinese_remainder_reconstruction_univariate(hp, hq, p, q) >>> hpq 2*x**3 + 3*x**2 + 6*x + 5 >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True rr*T symmetric)rrgensrrangercoeff strip_zero)r-hqr"qnxhpqis r,_chinese_remainder_reconstruction_univariater=[sl A  QA ',C 1Q3ZZUUA!Q$!Q$ @DQQQRSTQD NN Jrc6|j|jkr|jjjsJt||}||S|j}|\}}|\}}|j||}t ||}|dkr:|||||z|||zfS|j|j|j}d} d} t| } || zdkrt| } || zdk| | } | | } t| | | } | }||kr||krd} |}| | | } | dkr| } | }t| || | }| | z} ||ks|}||}||\}}||\}}|sZ|sX|jdkr| }||}|||z}|||z}|||fS)a Computes the GCD of two polynomials in `\mathbb{Z}[x]` using a modular algorithm. The algorithm computes the GCD of two univariate integer polynomials `f` and `g` by computing the GCD in `\mathbb{Z}_p[x]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. Trial division is only made for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement univariate integer polynomial g : PolyElement univariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_univariate >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**5 - 1 >>> g = x - 1 >>> h, cff, cfg = modgcd_univariate(f, g) >>> h, cff, cfg (x - 1, x**4 + x**3 + x**2 + x + 1, 1) >>> cff * h == f True >>> cfg * h == g True >>> f = 6*x**2 - 6 >>> g = 2*x**2 + 4*x + 2 >>> h, cff, cfg = modgcd_univariate(f, g) >>> h, cff, cfg (2*x + 2, 3*x - 3, x + 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ Nrr*)rris_ZZr primitiver+r/rrrrr(rr= quo_groundcontentdiv)rrresultrcfcgchboundr,mr"r r!r-r.hlastmhmhfquofremgquogremcffcfgs rmodgcd_univariaterSsF 6QV 3 3333 !Q  F  6D KKMMEB KKMMEB R B $Q * *E zHtBxxbBh//bBh1G1GGG KOOAD!$ ' 'E A A' aLLai1n ! Aai1n ^^A   ^^A   RQ    5=   U] AE  ]]5 ! ! . .q 1 1 6 AF  9"fa K K QV| F  MM"**,, ' 'UU1XX dUU1XX d D tax S R  A//"(++C//"(++Cc3; O'rc B|j}|j}|j}i}|D]7\}}|dd|vr i||dd<|||dd|d<8g}t |D]#}t |t|||||}$||j |dz } | | |} | | | |fS)a Compute the content and the primitive part of a polynomial in `\mathbb{Z}_p[x_0, \ldots, x_{k-2}, y] \cong \mathbb{Z}_p[y][x_0, \ldots, x_{k-2}]`. Parameters ========== f : PolyElement integer polynomial in `\mathbb{Z}_p[x0, \ldots, x{k-2}, y]` p : Integer modulus of `f` Returns ======= contf : PolyElement integer polynomial in `\mathbb{Z}_p[y]`, content of `f` ppf : PolyElement primitive part of `f`, i.e. `\frac{f}{contf}` Examples ======== >>> from sympy.polys.modulargcd import _primitive >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> p = 3 >>> f = x**2*y**2 + x**2*y - y**2 - y >>> _primitive(f, p) (y**2 + y, x**2 - 1) >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x*y*z - y**2*z**2 >>> _primitive(f, p) (z, x*y - y**2*z) Nr*symbols)rrngens itertermsitervaluesrr clonerW from_denserquoset_ring) rr"rr#kcoeffsmonomr5contyringcontfs r _primitiverfs(R 6D +C A F .. u ":V # $!#F5": (-uSbSz59%% Dfmmoo&&AAdL377C@@ JJt|AaC0J 1 1E   T " " / / 2 2E !%%t,,-- --rc|jj}d|dz z}|D]}|dd|kr |dd}|S)a Compute the degree of a multivariate polynomial `f \in K[x_0, \ldots, x_{k-2}, y] \cong K[y][x_0, \ldots, x_{k-2}]`. Parameters ========== f : PolyElement polynomial in `K[x_0, \ldots, x_{k-2}, y]` Returns ======= degf : Integer tuple degree of `f` in `x_0, \ldots, x_{k-2}` Examples ======== >>> from sympy.polys.modulargcd import _deg >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _deg(f) (2,) >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _deg(f) (2, 2) >>> f = x*y*z - y**2*z**2 >>> _deg(f) (1, 1) )rr*NrU)rrX itermonoms)rr`degfrbs r_degrjZsZP  A 1Q3>> from sympy.polys.modulargcd import _LC >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _LC(f) y**2 + y >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _LC(f) 1 >>> f = x*y*z - y**2*z**2 >>> _LC(f) z r*rVrNrU)rrXr\rWr3rjrrY) rrr`rdyrilcfrbr5s r_LCrnsP 6D A JJt|AaC0J 1 1E 1 A 77D *C && u ":  & 5E"I% %C Jrc|j}|j}|D],\}}||f|d|z||dzdz}|||<-|S)zS Make the variable `x_i` the leading one in a multivariate polynomial `f`. Nr*)rrrY)rr<rfswaprbr5 monomswaps r_swaprrsh 6D IE !! u1XK%)+eAaCDDk9  i LrcF|j}|j|j|j}|jt |djt |dj}||z}d}t |}||zdkrt |}||zdk||}||}t||\} }t||\} }t| | |} | } tt|t||} t|D]}| d||zs| d||}| d||}t|||}| }|| fcSt| | | fS)a Compute upper degree bounds for the GCD of two bivariate integer polynomials `f` and `g`. The GCD is viewed as a polynomial in `\mathbb{Z}[y][x]` and the function returns an upper bound for its degree and one for the degree of its content. This is done by choosing a suitable prime `p` and computing the GCD of the contents of `f \; \mathrm{mod} \, p` and `g \; \mathrm{mod} \, p`. The choice of `p` guarantees that the degree of the content in `\mathbb{Z}_p[y]` is greater than or equal to the degree in `\mathbb{Z}[y]`. To obtain the degree bound in the variable `x`, the polynomials are evaluated at `y = a` for a suitable `a \in \mathbb{Z}_p` and then their GCD in `\mathbb{Z}_p[x]` is computed. If no such `a` exists, i.e. the degree in `\mathbb{Z}_p[x]` is always smaller than the one in `\mathbb{Z}[y][x]`, then the bound is set to the minimum of the degrees of `f` and `g` in `x`. Parameters ========== f : PolyElement bivariate integer polynomial g : PolyElement bivariate integer polynomial Returns ======= xbound : Integer upper bound for the degree of the GCD of the polynomials `f` and `g` in the variable `x` ycontbound : Integer upper bound for the degree of the content of the GCD of the polynomials `f` and `g` in the variable `y` References ========== 1. [Monagan00]_ r*r)rrr+rrrrrrfr(rrnr4evaluatemin)rrrgamma1gamma2 badprimesr"r r!contfpcontgpconthp ycontbounddeltaafpagpahpaxbounds r_degree_bound_bivariatersT 6D [__QT14 ( (F [__U1a[[^U1a[[^ < >> from sympy.polys.modulargcd import _chinese_remainder_reconstruction_multivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> p = 3 >>> q = 5 >>> hp = x**3*y - x**2 - 1 >>> hq = -x**3*y - 2*x*y**2 + 2 >>> hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) >>> hpq 4*x**3*y + 5*x**2 + 3*x*y**2 + 2 >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True >>> R, x, y, z = ring("x, y, z", ZZ) >>> p = 6 >>> q = 5 >>> hp = 3*x**4 - y**3*z + z >>> hq = -2*x**4 + z >>> hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) >>> hpq 3*x**4 + 5*y**3*z + z >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True c:t||g||gddS)NTr1rr)cpcqr"r8s rcrt_z<_chinese_remainder_reconstruction_multivariate..crt_ks$1vBx4888; ;r) setmonoms intersectiondifference_updaterrr isinstancer._chinese_remainder_reconstruction_multivariate) r-r7r"r8hpmonomshqmonomsrrr;rrbs rrrs@P299;;H299;;H  " "8 , ,F v&&& v&&& 7> D ',C"'..11<= < < <66T"U)RY155E 11T"U)T1a00E 11T$5 1a00E JrFc|j}|r|jj}|jj|}n|j}|j|}t||D]u\} } |j} |j} |D]} | | kr | || z z} | | | z z} || |} | | }|| ||zz }v||S)a Reconstruct a polynomial `h_p` in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` from a list of evaluation points in `\mathbb{Z}_p` and a list of polynomials in `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, which are the images of `h_p` evaluated in the variable `x_i`. It is also possible to reconstruct a parameter of the ground domain, i.e. if `h_p` is a polynomial over `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`. In this case, one has to set ``ground=True``. Parameters ========== evalpoints : list of Integer objects list of evaluation points in `\mathbb{Z}_p` hpeval : list of PolyElement objects list of polynomials in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, images of `h_p` evaluated in the variable `x_i` ring : PolyRing `h_p` will be an element of this ring i : Integer index of the variable which has to be reconstructed p : Integer prime number, modulus of `h_p` ground : Boolean indicates whether `x_i` is in the ground domain, default is ``False`` Returns ======= hp : PolyElement interpolated polynomial in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` ) rrr3ziprrrr_r) evalpointshpevalrr<r"groundr-rrlr~rnumerdenombr5s r_interpolate_multivariaterxsN B # K Q  IaLj&)) ) )3   AAv  QUNE QUNEE eQ''  '' cll4  5(( ??1  rc |j|jkr|jjjsJt||}||S|j}|\}}|\}}|j||}t ||\}}||cxkrdkr=nn:|||||z|||zfSt|d} t|d} | } | } t | | \} }| |cxkrdkr=nn:|||||z|||zfS|j|j |j }|j| j | j }||z}d}d} t|}||zdkrt|}||zdk| |}| |}t||\}}t||\}}t|||}| }||kr||krd}|}tt|t||}| }| }| }t!| |z | |z | |z |zdz}||krGd}g} g}!d}"t#|D] }#|d|#}$|$|zs|d|# |}%|d|# |}&t|%|&|}'|' }(|(|kr|(|krd}|(}d}"na|'|$ |}'| |#|!|'|dz }||krn |"rn||krvt)| |!|d|})t|)|d})|)||z})|) d}*|*| kr|*| krd}|*} |)| |})|dkr|}|)}+t-|)|+||},||z}|,|+ks|,}+5|,|,}-||-\}.}/||-\}0}1|/sZ|1sX|-j dkr| }|-|}-|.||z}2|0||z}3|-|2|3fS)a! Computes the GCD of two polynomials in `\mathbb{Z}[x, y]` using a modular algorithm. The algorithm computes the GCD of two bivariate integer polynomials `f` and `g` by calculating the GCD in `\mathbb{Z}_p[x, y]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. To compute the bivariate GCD over `\mathbb{Z}_p`, the polynomials `f \; \mathrm{mod} \, p` and `g \; \mathrm{mod} \, p` are evaluated at `y = a` for certain `a \in \mathbb{Z}_p` and then their univariate GCD in `\mathbb{Z}_p[x]` is computed. Interpolating those yields the bivariate GCD in `\mathbb{Z}_p[x, y]`. To verify the result in `\mathbb{Z}[x, y]`, trial division is done, but only for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement bivariate integer polynomial g : PolyElement bivariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_bivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 - y**2 >>> g = x**2 + 2*x*y + y**2 >>> h, cff, cfg = modgcd_bivariate(f, g) >>> h, cff, cfg (x + y, x - y, x + y) >>> cff * h == f True >>> cfg * h == g True >>> f = x**2*y - x**2 - 4*y + 4 >>> g = x + 2 >>> h, cff, cfg = modgcd_bivariate(f, g) >>> h, cff, cfg (x + 2, x*y - x - 2*y + 2, 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ Nrr*TF)rrr?rr@r+rrrrrrrrrfr(rnrur4rtappendrr_rrArBrC)4rrrDrrErFrGrr|rpgswapdegyfdegygybound xcontboundrvrwrxrIr"r r!ryrzr{ degconthpr} degcontfp degcontgpdegdeltaNr9rrunluckyr~deltaarrrdeghpar-degyhprJrKrLrMrNrOrPrQrRs4 rmodgcd_bivariatersKR 6QV 3 3333 !Q  F  6D KKMMEB KKMMEB R B0A66FJ HHHHqHHHHHtBxxbBh//bBh1G1GGG !QKKE !QKKE LLNNE LLNNE0>>FJ HHHHqHHHHHtBxxbBh//bBh1G1GGG[__QT14 ( (F [__UXux 0 0FI A Ag aLL!mq  ! A!mq  ^^A   ^^A  A&& A&& ++MMOO z !    # A"J BR!,,MMOO MMOO <<>>  !59#4 Z ( * , ,./ 0 q5    q  A^^Aq))FA: ++a##0033C++a##0033C#sA&&CZZ\\F & ..((55a88C   a MM#    FAAv      q5   &z64A F F A  q ! &//$'' '1 F?   F? AF  ]]6 " " / / 2 2 6 AF  ;B1 M M QV| F  MM"**,, ' 'UU1XX dUU1XX d D tax S R  A//"(++C//"(++Cc3; Ogrc |j}|j}|dkr`t||||}|}||dkrdS||dkr ||d<t |S||dz } ||dz } t ||\} }t ||\} }t| | |} | }| }| }|||dz krdS|||dz kr|||dz <t t|}t|}t|||}|}t|dz D]L}|ttt||tt|||z}M|}t| |z | |z ||dz ||dz z |zdz}||krdSd}d}g}g}tt|}|rtj |dd}|||d||zsM|d||z}||dz ||}||dz ||} t!|| |||}!|!|dz }||krdS|!jr*| ||}|S|!||}!||||!|dz }||krt+||||dz |}t ||d| |z}||dz }"|"||dz krdS|"||dz kr|"||dz <t |S|dS)a Compute the GCD of two polynomials in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`. The algorithm reduces the problem step by step by evaluating the polynomials `f` and `g` at `x_{k-1} = a` for suitable `a \in \mathbb{Z}_p` and then calls itself recursively to compute the GCD in `\mathbb{Z}_p[x_0, \ldots, x_{k-2}]`. If these recursive calls are successful for enough evaluation points, the GCD in `k` variables is interpolated, otherwise the algorithm returns ``None``. Every time a GCD or a content is computed, their degrees are compared with the bounds. If a degree greater then the bound is encountered, then the current call returns ``None`` and a new evaluation point has to be chosen. If at some point the degree is smaller, the correspondent bound is updated and the algorithm fails. Parameters ========== f : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_p` g : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_p` p : Integer prime number, modulus of `f` and `g` degbound : list of Integer objects ``degbound[i]`` is an upper bound for the degree of the GCD of `f` and `g` in the variable `x_i` contbound : list of Integer objects ``contbound[i]`` is an upper bound for the degree of the content of the GCD in `\mathbb{Z}_p[x_i][x_0, \ldots, x_{i-1}]`, ``contbound[0]`` is not used can therefore be chosen arbitrarily. Returns ======= h : PolyElement GCD of the polynomials `f` and `g` or ``None`` References ========== 1. [Monagan00]_ 2. [Brown71]_ r*rN)rrXr(rrr rfrnr4rrrulistrandomsampleremovert_modgcd_multivariate_p is_groundr_rrr)#rrr"degbound contboundrr`rLdeghrrrecontgconthdegcontfdegcontgdegconthrmlcgr}evaltestr<rrr9drhevalpointsr~rfagahadegyhs# rrrss` 6D AAv  Aq!   ) )! , ,xxzz (1+  4 (1+  #HQK" " HHQqSMME HHQqSMME!QHE1!QHE1 E5! $ $E||~~H||~~H||~~H)AaC. t)AaC. ! !A# a&&C a&&C Ca EH 1Q3ZZCCGCa ,,c%1++.>.>BBB||~~H EH eh. QqSMIacN *X 5 7 79: ;A 1ut A AJ E %((^^F + M&! $ $Q ' a  A&&*  1%%) ZZ!Q   , ,Q / / ZZ!Q   , ,Q / /$BAx C C   FA1u t  < t$$11!44AH ]]6 " " / / 2 2! R Q 6 )*eT1Q3JJA1a  #ennT&:&::AHHQqSMMEx!}$ tx!}$ ' %1 &&HW +Z 4rc |j|jkr|jjjsJt||}||S|j}|j}|\}}|\}}|j||}|j|j|j}|jj} t|D]F} | |jt|| jt|| jz} Gdt| | D} t| } d} d} t|}| |zdkrt|}| |zdk||}||} t!|||| | }n#t"$rd} Y~wxYw||||}| dkr|} |}t'|||| }| |z} ||ks|}|d}||\}}||\}}|sZ|sX|jdkr| }||}|||z}|||z}|||fS)a Compute the GCD of two polynomials in `\mathbb{Z}[x_0, \ldots, x_{k-1}]` using a modular algorithm. The algorithm computes the GCD of two multivariate integer polynomials `f` and `g` by calculating the GCD in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. To compute the multivariate GCD over `\mathbb{Z}_p` the recursive subroutine :func:`_modgcd_multivariate_p` is used. To verify the result in `\mathbb{Z}[x_0, \ldots, x_{k-1}]`, trial division is done, but only for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement multivariate integer polynomial g : PolyElement multivariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_multivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 - y**2 >>> g = x**2 + 2*x*y + y**2 >>> h, cff, cfg = modgcd_multivariate(f, g) >>> h, cff, cfg (x + y, x - y, x + y) >>> cff * h == f True >>> cfg * h == g True >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x*z**2 - y*z**2 >>> g = x**2*z + z >>> h, cff, cfg = modgcd_multivariate(f, g) >>> h, cff, cfg (z, x*z - y*z, x**2 + 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ 2. [Brown71]_ See also ======== _modgcd_multivariate_p Nc4g|]\}}t||S)ru).0fdeggdegs r z'modgcd_multivariate..s$PPPJD$D$PPPrr*Tr)rrr?rrXr@r+rrr4rrrdegreesrrrrr rrrC)rrrDrr`rErFrGr,rxr<rrrIr"r r!r-rJrKrLrMrNrOrPrQrRs rmodgcd_multivariater&s\ 6QV 3 3333 !Q  F  6D A KKMMEB KKMMEB R B KOOAD!$ ' 'E I 1XXEET[__U1a[[^U1a[[^DDD PP#aiikk199;;2O2OPPPHXI A A( aLL!mq  ! A!mq  ^^A   ^^A   'B8YGGBB   A H     ]]5 ! ! . .q 1 1 6 AF  ;B1 M M QV| F  LLNN1 UU1XX dUU1XX d D tax S R  A//"(++C//"(++Cc3; Q(sG G&%G&c|j}t||||j\}}||||fS)z_ Compute `\frac f g` modulo `p` for two univariate polynomials over `\mathbb Z_p`. )rr to_denserr])rrr"rdensequodenserems r_gf_divrsX 6D ajjllAt{KKHh ??8 $ $dooh&?&? ??rc(|j}|j}|}|dz}||z dz }||j} }||j} } | |krit || |d} | || | zz |} }| | | | zz |} } | |ki| | }} ||kst|||dkrdS|j}|dkrf| ||}| ||} | ||}| }|| ||z S)a  Reconstruct a rational function `\frac a b` in `\mathbb Z_p(t)` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are polynomials in `\mathbb Z_p[t]` and `m` has positive degree. The algorithm is based on the Euclidean Algorithm. In general, `m` is not irreducible, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : PolyElement univariate polynomial in `\mathbb Z[t]` p : Integer prime number m : PolyElement modulus, not necessarily irreducible Returns ======= frac : FracElement either `\frac a b` in `\mathbb Z(t)` or ``None`` References ========== 1. [Hoeij04]_ r*rN) rrrrrrrr(rrrto_field)cr"rIrrMrDr0s0r1s1r^r~rlcr&fields r!_rational_function_reconstructionrsJ 6D [F  A QA A A  B B ))++/3b"a  #b3r6k//22Bb3r6k//22B ))++/3 rqAxxzzA~Aq))Q.t B Qw0 b!$$ LL   , ,Q / / LL   , ,Q / / MMOOE 588eeAhh rc6|j}|D]|\}}|dkrt|||}|sdSnU|jj}||D]!\} } t| ||} | sdS| || <"|||<}|S)a Reconstruct every coefficient `c_h` of a polynomial `h` in `\mathbb Z_p(t_k)[t_1, \ldots, t_{k-1}][x, z]` from the corresponding coefficient `c_{h_m}` of a polynomial `h_m` in `\mathbb Z_p[t_1, \ldots, t_k][x, z] \cong \mathbb Z_p[t_k][t_1, \ldots, t_{k-1}][x, z]` such that .. math:: c_{h_m} = c_h \; \mathrm{mod} \, m, where `m \in \mathbb Z_p[t]`. The reconstruction is based on the Euclidean Algorithm. In general, `m` is not irreducible, so it is possible that this fails for some coefficient. In that case ``None`` is returned. Parameters ========== hm : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][x, z]` p : Integer prime number, modulus of `\mathbb Z_p` m : PolyElement modulus, polynomial in `\mathbb Z[t]`, not necessarily irreducible ring : PolyRing `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]`, `h` will be an element of this ring k : Integer index of the parameter `t_k` which will be reconstructed Returns ======= h : PolyElement reconstructed polynomial in `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]` or ``None`` See also ======== _rational_function_reconstruction rN)rrYrrdrop_to_ground) rKr"rIrr`rLrbr5coeffhmonrrGs r$_rational_reconstruction_func_coeffsrs\ A  u 6 !6uaCCF tt [%F..q11;;== ! !Q6q!Q?? 444 s % Hrc |j}t||||j\}}}||||||fS)z Extended Euclidean Algorithm for two univariate polynomials over `\mathbb Z_p`. Returns polynomials `s, t` and `h`, such that `h` is the GCD of `f` and `g` and `sf + tg = h \; \mathrm{mod} \, p`. )rr rrr])rrr"rstrLs r _gf_gcdexrKsg 6Dqzz||QZZ\\1dkBBGAq! ??1  tq114??13E3E EErc|j}||}||}||||g|S)a Compute the reduced representation of a polynomial `f` in `\mathbb Z_p[z] / (\check m_{\alpha}(z))[x]` Parameters ========== f : PolyElement polynomial in `\mathbb Z[x, z]` minpoly : PolyElement polynomial `\check m_{\alpha} \in \mathbb Z[z]`, not necessarily irreducible p : Integer prime number, modulus of `\mathbb Z_p` Returns ======= ftrunc : PolyElement polynomial in `\mathbb Z[x, z]`, reduced modulo `\check m_{\alpha}(z)` and `p` )rr_ ground_newrr$)rminpolyr"rp_s r_truncrYsa0 6Dt$$G   B >>!   '2 / / < >t D DF !f*gq ) ))rc|j}||jd|jdf}||}|}|}|}|d} t ||} |j} |r9||kr2t ||} || z|||z df| zz }|r||}|d}|r|| krt |||} | | |d|| z f| zz }|r||}|d}|r|| k|}|r||k2|S)a= Check if `h` divides `f` in `\mathbb K[t_1, \ldots, t_k][z]/(m_{\alpha}(z))`, where `\mathbb K` is either `\mathbb Q` or `\mathbb Z_p`. This algorithm is based on pseudo division and does not use any fractions. By default `\mathbb K` is `\mathbb Q`, if a prime number `p` is given, `\mathbb Z_p` is chosen instead. Parameters ========== f, h : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement polynomial `m_{\alpha}(z)` in `\mathbb Z[t_1, \ldots, t_k][z]` p : Integer or None if `p` is given, `\mathbb K` is set to `\mathbb Z_p` instead of `\mathbb Q`, default is ``None`` Returns ======= rem : PolyElement remainder of `\frac f h` References ========== .. [1] [Hoeij02]_ r*rrV) rr\rWr_rrnrrrr) rrLrr"rzxringr$r'rdegmlchlcmlcrems r_trial_divisionrsB 6D ZZa$,q/ BZ C CFt$$G C ZZ\\F 88::D >>!  D a&&//$  C *C &D.C!!$''#g Vd]A$677==  &""1%%CA #fn # V,,--66t<\}} }!|!|kr2|||| |||z z}?||}| |||||| dz } t=||||dz |d}"t?|"||||dz }"|"|dkr|jj }#|#jj}$|"!D]Z}|#j"tG|$$|j%$||#j}$[n|jjj }#|#jj}$|"!D]q}|!D]Z}%|#j"tG|$$|%j%$||#j}$[r|&|$|}$||"'|$(|}"tS||"||stS||"||s|"S|dS)a Compute the GCD of two polynomials `f` and `g` in `\mathbb Z_p(t_1, \ldots, t_k)[z]/(\check m_\alpha(z))[x]`. The algorithm reduces the problem step by step by evaluating the polynomials `f` and `g` at `t_k = a` for suitable `a \in \mathbb Z_p` and then calls itself recursively to compute the GCD in `\mathbb Z_p(t_1, \ldots, t_{k-1})[z]/(\check m_\alpha(z))[x]`. If these recursive calls are successful, the GCD over `k` variables is interpolated, otherwise the algorithm returns ``None``. After interpolation, Rational Function Reconstruction is used to obtain the correct coefficients. If this fails, a new evaluation point has to be chosen, otherwise the desired polynomial is obtained by clearing denominators. The result is verified with a fraction free trial division. Parameters ========== f, g : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][z]`, not necessarily irreducible p : Integer prime number, modulus of `\mathbb Z_p` Returns ======= h : PolyElement primitive associate in `\mathbb Z[t_1, \ldots, t_k][x, z]` of the GCD of the polynomials `f` and `g` or ``None``, coefficients are in `\left[ -\frac{p-1} 2, \frac{p-1} 2 \right]` References ========== 1. [Hoeij04]_ r*rrNT)r)*rrrrrXrrrr\rrrr4rrrrtrrr$_func_field_modgcd_prrYLMtupleget_ringrr3rrrrr itercoeffsr]r rr domain_newras_exprr)&rrrr"rrr`qdomainqringr9rr,r}rrLMlistrr~testgammaaminpolyarrrrrbr5 evalpoints_aheval_arIrrhbLMhbrLr#denrs& rrrsT 6D [F&.))6 L#Aq'1555AvG+&&((+,,QU33--w~':'C'C'E'E-FF JJgJ & &E A A KKNNT[[^^ +E JEJ E F %((^^F Z M&! $ $Q ' a 6 ?>>!A#q))A-2DD>>!A#q))66q99Q>D   !%1a00#GQqS!44 ::xQ0 1 1Q 6   a1a ( ( a1a ( (""b(A 6 6   FA1u t  7 Iiikk]aS!A#Y & q5 & "  & & u8r!u$&E"QRR&MM)A&"XBqrrFs $ 6 3 %%''+AA #,,..2A FKNz5&99  KAr4rz ##A&&&r"""a!e  NN1  ! R b Q &lGT1Q3RV W W W 1Aq%1 E E    6 ,,$C(,C ( (h))&AUAUAWAW3:+'+'(( ( ,%+C(,C , ,))++,,A(--fS\\^^QWEUEUEWEWsz/+/+,,CC,s//2233 Dc""**,, - - : :1 = =q!Wa00 AwXY9Z9Z Hu Zx 4rc^|dkr||z }||j}}||j}}t|dz }||kr||z}||||zz }}||||zz }}||kt||krdS|dkr| | } } n |dkr||} } ndS|} | | | | z S)a Reconstruct a rational number `\frac a b` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are integers. The algorithm is based on the Euclidean Algorithm. In general, `m` is not a prime number, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : Integer `c = \frac a b \; \mathrm{mod} \, m` m : Integer modulus, not necessarily prime domain : IntegerRing `a, b, c` are elements of ``domain`` Returns ======= frac : Rational either `\frac a b` in `\mathbb Q` or ``None`` References ========== 1. [Wang81]_ rrN)rrrabs get_field) rrIrrrrrrHr^r~rrs r _integer_rational_reconstructionrsH 1u Q  B  B QKKE +!BhR#b&[BR#b&[B +!  2ww%t AvsRC1 a21t     E 588eeAhh rc|j}t|jtrt}|jj}nt }|jj}|D]\}}||||}|sdS|||<|S)a Reconstruct every rational coefficient `c_h` of a polynomial `h` in `\mathbb Q[t_1, \ldots, t_k][x, z]` from the corresponding integer coefficient `c_{h_m}` of a polynomial `h_m` in `\mathbb Z[t_1, \ldots, t_k][x, z]` such that .. math:: c_{h_m} = c_h \; \mathrm{mod} \, m, where `m \in \mathbb Z`. The reconstruction is based on the Euclidean Algorithm. In general, `m` is not a prime number, so it is possible that this fails for some coefficient. In that case ``None`` is returned. Parameters ========== hm : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][x, z]` m : Integer modulus, not necessarily prime ring : PolyRing `\mathbb Q[t_1, \ldots, t_k][x, z]`, `h` will be an element of this ring Returns ======= h : PolyElement reconstructed polynomial in `\mathbb Q[t_1, \ldots, t_k][x, z]` or ``None`` See also ======== _integer_rational_reconstruction N)rrrr#_rational_reconstruction_int_coeffsrrrY) rKrIrrLreconstructionrrbr5rs rrrsR A$+~.. <!9  uq&11 44% Hrc|j}|j}t|trP|j}|j|j}||}n/d}||j}|\}}|\} }||||z} |j } d} g} g}g} t| } | | dkr*|dkr | | zdk}n| | dk}|rV| | }| | }| | }t|||| }||dkr|j S|gdg|zz}|dkrX|D]C\}}|d|dkr,|jt#|ddkr |j|dd<D|}| }t%| ||D]#\}}}||krt'||||}||z}$| | ||||t+|||}||dkr|d}nk|jj }|D]5}|j||d}6||}||}|d}t7||||s&t7|| ||s|S)a Compute the GCD of two polynomials in `\mathbb Q(t_1, \ldots, t_k)[z]/(m_{\alpha}(z))[x]` using a modular algorithm. The algorithm computes the GCD of two polynomials `f` and `g` by calculating the GCD in `\mathbb Z_p(t_1, \ldots, t_k)[z] / (\check m_{\alpha}(z))[x]` for suitable primes `p` and the primitive associate `\check m_{\alpha}(z)` of `m_{\alpha}(z)`. Then the coefficients are reconstructed with the Chinese Remainder Theorem and Rational Reconstruction. To compute the GCD over `\mathbb Z_p(t_1, \ldots, t_k)[z] / (\check m_{\alpha})[x]`, the recursive subroutine ``_func_field_modgcd_p`` is used. To verify the result in `\mathbb Q(t_1, \ldots, t_k)[z] / (m_{\alpha}(z))[x]`, a fraction free trial division is used. Parameters ========== f, g : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement irreducible polynomial in `\mathbb Z[t_1, \ldots, t_k][z]` Returns ======= h : PolyElement the primitive associate in `\mathbb Z[t_1, \ldots, t_k][x, z]` of the GCD of `f` and `g` Examples ======== >>> from sympy.polys.modulargcd import _func_field_modgcd_m >>> from sympy.polys import ring, ZZ >>> R, x, z = ring('x, z', ZZ) >>> minpoly = (z**2 - 2).drop(0) >>> f = x**2 + 2*x*z + 2 >>> g = x + z >>> _func_field_modgcd_m(f, g, minpoly) x + z >>> D, t = ring('t', ZZ) >>> R, x, z = ring('x, z', D) >>> minpoly = (z**2-3).drop(0) >>> f = x**2 + (t + 1)*x*z + 3*t >>> g = x*z + 3*t >>> _func_field_modgcd_m(f, g, minpoly) x + t*z References ========== 1. [Hoeij04]_ See also ======== _func_field_modgcd_p rrr*TN)rrrrrXr\rr@rrrrrrrrYrrrrrr clear_denomsrrrr_r)rrrrrr`QQdomainQQringrErFr,r}r"primeshplistrrr r!minpolypr-rrbr5rKrIr8r7LMhqrLrs r_func_field_modgcd_mr&sD 6D [F&.))< L;$$FM,C,C,E,E$FF8,, 4;#8#8#:#:;; KKMMEB KKMMEB KKNNT[[^^ +E JE A F F F? aLL   a A %   6 0AINDD&&q))Q.D    ^^A   ^^A  ''** !"b(A 6 6    7 8Oiikk]aSU " q5 & "  & & u8r!u$&E"QRR&MM)A&"XBqrrF  vvv66  KAr4rz CBAqQQQ a b b 0Q ? ?    6 #!!!$AA-#C F Fm''U-?-?-A-A!-DEE c""A JJt   KKMM!  R 0 0!W==  ALL,,a 9 9 H?rc|j}t|jtr |jj}n|j}|j}|D])}|jD]}|r|||j} *| D]\}}|j}|jj}t|jtr| |dd}t|} t| D]e} || r[||| |z|z}|d| | z dz f|vr|||d| | z dz f<G||d| | z dz fxx|z cc<f|S)a Compute an associate of a polynomial `f \in \mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` in `\mathbb Z[x_1, \ldots, x_{n-1}][z] / (\check m_{\alpha}(z))[x_0]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` ring : PolyRing `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` Returns ======= f_ : PolyElement associate of `f` in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` r*Nr) rrrrrrrepr denominatorrYrlenr4) rrf_rrr5rrbrIr9r<s r _to_ZZ_polyr$s2 B$+~..# *C55 5 5A 5jjam44 5 // u  KO dk> 2 2 ' E!""I&&A JJq / /AQx /F58c>**Q.!Hac!e$B./,-Ba!A#a%())a!A#a%()))Q.))) / Irc@|j}|j}t|jjtr|D]q\}}|D]W\}}|df|z}|||gdg|dzz} ||vr| ||<G||xx| z cc<Xrni|D]T\}}|df}|||gdg|dzz} ||vr| ||<D||xx| z cc<U|S)ar Convert a polynomial `f \in \mathbb Z[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha}(z))[x_0]` to a polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` ring : PolyRing `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= f_ : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` rr*)rrrrrrY) rrrr#rbr5rcoefrIrs r _to_ANP_polyr's]0[F B!&-00KKMM  LE5"__..   T1XK#%FFMM$//0A3uQx<?@@B;BqEEqEEEQJEEEE  KKMM  LE5q A e,,-E!H <==A{ 11  Ircx|j}|D]\}}||||<|S)zo Change representation of the minimal polynomial from ``DMP`` to ``PolyElement`` for a given ring. )rtermsr)rrminpoly_rbr5s r_minpoly_from_denser+/sB yH -- u++e,, Orcz|j}|jtd|j}|jj}||}|j}|D])}t||d}||j kr||fcS*|| | |fS)z Compute the content in `x_0` and the primitive part of a polynomial `f` in `\mathbb Q(\alpha)[x_0, x_1, \ldots, x_{n-1}] \cong \mathbb Q(\alpha)[x_1, \ldots, x_{n-1}][x_0]`. r*r) rrr4rXrrrrfunc_field_modgcdrr^r_)rfringrr#r#rcr5s r_primitive_in_x0r/<s FE 5 q%+!6!6 7D + C aiikk  B 8D u--a0 37? 7NNN  t}}U++,, ,,rc|j}|j}|j}||jkr|jsJt ||}||St d}||j|fz|j}|dkrut||}t||} | d |j j } t|| | } t| |} nt!|\} }t!|\} }t#| | d}|jt'd|}t||}t||} t)|j | d} t|| | } t| |} t!| \}} | ||z} || |z}|| |z}| | j} | || || fS)a Compute the GCD of two polynomials `f` and `g` in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` using a modular algorithm. The algorithm first computes the primitive associate `\check m_{\alpha}(z)` of the minimal polynomial `m_{\alpha}` in `\mathbb{Z}[z]` and the primitive associates of `f` and `g` in `\mathbb{Z}[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha})[x_0]`. Then it computes the GCD in `\mathbb Q(x_1, \ldots, x_{n-1})[z]/(m_{\alpha}(z))[x_0]`. This is done by calculating the GCD in `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem and Rational Reconstuction. The GCD over `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]` is computed with a recursive subroutine, which evaluates the polynomials at `x_{n-1} = a` for suitable evaluation points `a \in \mathbb Z_p` and then calls itself recursively until the ground domain does no longer contain any parameters. For `\mathbb{Z}_p[z]/(\check m_{\alpha}(z))[x_0]` the Euclidean Algorithm is used. The results of those recursive calls are then interpolated and Rational Function Reconstruction is used to obtain the correct coefficients. The results, both in `\mathbb Q(x_1, \ldots, x_{n-1})[z]/(m_{\alpha}(z))[x_0]` and `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]`, are verified by a fraction free trial division. Apart from the above GCD computation some GCDs in `\mathbb Q(\alpha)[x_1, \ldots, x_{n-1}]` have to be calculated, because treating the polynomials as univariate ones can result in a spurious content of the GCD. For this ``func_field_modgcd`` is called recursively. Parameters ========== f, g : PolyElement polynomials in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= h : PolyElement monic GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac f h` cfg : PolyElement cofactor of `g`, i.e. `\frac g h` Examples ======== >>> from sympy.polys.modulargcd import func_field_modgcd >>> from sympy.polys import AlgebraicField, QQ, ring >>> from sympy import sqrt >>> A = AlgebraicField(QQ, sqrt(2)) >>> R, x = ring('x', A) >>> f = x**2 - 2 >>> g = x + sqrt(2) >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == x + sqrt(2) True >>> cff * h == f True >>> cfg * h == g True >>> R, x, y = ring('x, y', A) >>> f = x**2 + 2*sqrt(2)*x*y + 2*y**2 >>> g = x + sqrt(2)*y >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == x + sqrt(2)*y True >>> cff * h == f True >>> cfg * h == g True >>> f = x + sqrt(2)*y >>> g = x + y >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == R.one True >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Hoeij04]_ Nz)rWrr*r)rrrX is_Algebraicrrr\rWrr$rr]modr rr'r/r-rr4r+r_rArr^)rrrrr9rDr1ZZringr#g_rrLcontx0fcontx0gcontx0hZZring_contx0h_s rr-r-Qs)P 6D [F A 16>1f1111 !Q  F   c A ZZ t 3FMrBs############%%%%%%......44444444444444333333 ,???.B>>>B~~~B:.:.:.z---`222j   H5H5H5V___D>>>>BQQQhVVVrPPPf@@@???DC C C L F F F@@@>5*5*5*pBBBBJ   cccL===@: : : zYYYx777t000f   ---*T!T!T!T!T!r